Ananatural transformations

Idea

Given two functors interpreted as anafunctors, the natural transformations and ananatural transformations between them are the same, so the term ‘ananatural’ is overkill; one only needs it to emphasise the ana-context and otherwise can just say ‘natural’. That is, a ‘natural transformation’ between anafunctors unambigously means an ananatural transformation.

Definitions

Given two categoriesCC and DD and two anafunctors F,G:C→DF, G\colon C \to D, let us interpret F,GF,G as spansC←F¯→DC \leftarrow \overline{F} \rightarrow D and C←G¯→DC \leftarrow \overline{G} \rightarrow D of strict functors (where each backwards-pointing arrow is strictly surjective and faithful; see the definition of anafunctor). Form the strict22-pullbackP≔F¯×CG¯P \coloneqq \overline{F} \times_C \overline{G} and consider the strict functors P→F¯→DP \to \overline{F} \to D and P→G¯→DP \to \overline{G} \to D. Then an ananatural transformation from FF to GG is simply a natural transformation between these two strict functors.

More explicitly, if F,GF,G are given by sets |F|,|G|{|F|}, {|G|} of specifications and additional maps (see the other definition of anafunctor), then an ananatural transformation from FF to GG consists of a coherent family of morphisms of DD indexed by the elements of |F||F| and |G||G| with common values in CC. That is:

for each object xx of CC, each FF-specification ss over xx, and each GG-specification tt over xx, we have a morphism

ηs,t(x):Fs(x)→Gt(x) \eta_{s,t}(x)\colon F_s(x) \to G_t(x)

in DD;

for each morphism f:x→yf\colon x \to y in CC, each pair of FF-specifications s,ts,t over x,yx,y, and each pair of GG-specifications u,vu,v over x,yx,y, the diagram

Of course, an ananatural isomorphism is an invertible ananatural transformation.

Composition

Just as natural transformations can be composed vertically to form the morphisms of a functor category, so ananatural transformations can be composed vertically to form an anafunctor category.

Just as natural transformations can also be whiskered by functors and composed horizontally to make a strict 2-categoryStrCatStr Cat of (strict) categories, (strict) functors and natural transformations, so ananatural transformations can also be whiskered by anafunctors and composed horizontally to make a bicategoryCatanaCat_{ana} of (strict) categories, anafunctors and (ana)natural transformations. Assuming the axiom of choice, CatanaCat_{ana} is equivalent to StrCatStr Cat; without choice (and internally), CatanaCat_{ana} has better properties than StrCatStr Cat and we will usually identify the former with Cat.